The following paradigm is taken from [1]}.
The \(\mathpzc {F}_\bullet \)  are linear spaces and \(\mathpzc {E}_\bullet =\mathpzc {F}_\bullet ^*\)  their duals.
In our context they are some \(\mathbb {R}^d\) , so the duality inner product \(\langle \,\mathpzc {f}\hspace{-1.0pt},\,\mathpzc {e}\hspace{0.5pt}\,\rangle =\mathpzc {f}\hspace{-1.0pt}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}\) .
In general \(\mathcal {Z}\)  is a manifold, with \(T_z\mathcal {Z}\)  its tangent space at the state \(z\in \mathcal {Z}\) , and \(T_z^*\mathcal {Z}\)  the corresponding co-tangent space.
In our context \(\mathcal {Z}\)  is some \(\mathbb {R}^d\) , so \(T_z\mathcal {Z}\)  and \(T_z^*\mathcal {Z}\)  can also be regarded as \(\mathbb {R}^d\) , again with duality inner product.
